The Either-Or Wall

Four lines of arithmetic prove that no perceptron — trained, hand-set, or inspired — can sort an either-or-but-not-both pattern.

Suppose a stall sells out when festival or weekend but not both. Code the cases: neither — no; festival alone — yes; weekend alone — yes; both — no. Now suppose knobs exist: balance b, weights f and w. The no at neither requires b at most zero. The yes at festival alone requires b plus f above zero. The yes at weekend alone requires b plus w above zero. The no at both requires b plus f plus w at most zero. Add the two middle lines: twice b, plus f, plus w, exceeds zero. But the last line caps f plus w at minus b, so the sum is at most b — and the first line caps b at zero. A number above zero and at most zero: no such number exists, so no such knobs exist. The wall is proof-backed, not a matter of more training.